hi any can help to provide notes for hypersonic simulation using lattice boltzzman

First of all, how much hypersonic do you need? Lattice Boltzmann is for Mach 0.1-0.2. However, I think, it will be interesting to you to read the paper of X.Shan - something like Lattice Boltzmann beyond Navier-Stokes. In fact you can make better approximation for your equilibrium distribution in Mach expansion. You need to do it in a proper way and the paper provides the procedure.

P.S. I don’t think that you can achieve the Mach number more than 1 - but it comes only from feelings So try it and let us know.

I am not sure to understand the presumed restriction of LB to the Mach range 0.1-0.2, which seems at the same time quite optimistic and too pessimistic. Too pessimistic, because the BGK model (unlike MRT) easily handles the limit towards a zero Mach number, as it is for example shown in P. Dellar, 2003[/url] (but what Alex possibly wanted to say, is that 0.1-0.2 is an estimate for the maximum possible Mach number in LB, not the overall range of applicability). Quite optimistic, because at a Mach number as high as 0.1, you are in danger of encountering substantial numerical errors. If you use BGK and try to find the solution to the incompressible Navier-Stokes equations, you need to keep the Mach number low in order to avoid compressibility effects. In [url=http://www.lbmethod.org/literature:books]the book by S. Succi, it is recommended as a general guideline to keep the velocity in lattice units close to a value of 0.02 to ensure reasonable accuracy (in practice, the right value to use depends of course on various parameters). If you are simulating a compressible fluid, you might be able to go to somewhat higher Mach number. But generally, the low-Mach number limitation stays valid, because the numerical model is a low Mach-number expansion of the Boltzmann equation. Also, you must be aware that BGK has various limitations for representing compressible flows, such as the inability to adjust the bulk viscosity or the speed of sound.

The search for a high Mach-number LB model is strongly connected to the search for a consistent thermo-hydrodynamic model which inherently represents thermal effects. A lot of progress has been made by using higher-order discretizations of the Boltzmann equation, as in the paper by Shan which Alex pointed out (X. Shan, 2006). Although these models do include thermal effects and can reach higher Mach number, I have never heard of any of these being able to model supersonic flows.

An inherent limitation with LB is that the velocity, as measured in lattice units, cannot be larger than 1. Indeed (u>1) would mean that the fluid travels faster than one lattice site per time step, which is technically impossible with a nearest-neighbor dynamics. You can of course push this value a bit by using a larger neighborhood, but generally speaking, a limitation of u being of order 1 or smaller always persists. As in most LB models the speed of sound is a constant of order 1, this limitation also implies that one cannot reach a supersonic regime.

A workaround to reach the supersonic regime anyway is to say, OK, if we can’t increase u, let’s lower the speed of sound. This can be done for example by introducing an attractive force term, similar to the interaction term in the Shan/Chen model for multicomponent fluids. Such an approach is presented in H. Yu, 2000. I don’t really know how this model compares with classical CFD models for supersonic flows, but it is the only suggestion I have on this topic.